Remark 3.4  Let \(L\) , \(R\)  be transverse Dirac structures on \(M\) . As seen in Remark REF , almost Dirac structures transverse to \(R\)  are in bijection with elements of \(\Gamma \big ({\wedge }^2L^*\big )\) . We now recall a result of Liu–Weinstein–Xu [1]} establishing when such an almost Dirac structure is Dirac. Recall that every Dirac structure, with the restricted Dorfman bracket and anchor, is a Lie algebroid. Since \(L\)  is a Lie algebroid, it induces a differential \(d_L\)  on \(\Gamma (\wedge L^*)\) . FurtherThe Lie algebroid structures on \(L\)  and \(L^*\)  are compatible in the sense that the pair \((L,L^*)\)  forms a Lie bialgebroid., since \(L^*\cong R\)  is a Lie algebroid, it induces a graded Lie bracket \([\cdot ,\cdot ]_{L^*}\)  on \(\Gamma (\wedge L^*)[1]\) . Together with \(d_L\)  and \([\cdot ,\cdot ]_{L^*}\) , the graded vector space \(\Gamma (\wedge L^*)[1]\)  becomes a differential graded Lie algebra. The main result of [1]} is: for all \(\varepsilon \in \Gamma \big ({\wedge }^2L^*\big )\) , the graph \(L_{\varepsilon }=\lbrace v+\iota _v\varepsilon \colon v\in L\rbrace \)  is a Dirac structure iff \(\varepsilon \)  satisfies the Maurer–Cartan equation, that is
\( d_L\varepsilon + \tfrac{1}{2}[\varepsilon ,\varepsilon ]_{L^*} =0.\) 

